# Relationship between population density and population movement in inhabitable lands

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## Abstract

Population density that indicates a critical point in population growth is able to estimate from the relationship between population density and population movement. In the case of adopting usual definition of population density, the critical point depends on how to divide the space into unit of measurement. We find proper definition of population density. There are both inhabitable and uninhabitable areas. For population density adjusted by the size of the inhabitable area without agricultural land, the critical population density does not depend on the unit of measurement. We observe this statistical characteristic from detailed Japanese census data and land use data.

## Keywords

Demographic Census Econophysics Inhabitable land## JEL Classification

J100 C130 R120## 1 Introduction

Population is one of the most basic indices for estimating human statistics. The relationship between population density and various values has been investigated by several researchers. The correlation between population density and GDP was reported in Sutton et al. (2007) and Pan et al. (2013). The relationship between urban density and transport-related energy consumption was reported in Newman and Kenworthy (1988); Lefèvre (2009). The correlation between population density and crime was investigated in Hanley et al. (2016). A model of urban productivity with the agglomeration effect of population density was estimated in Abel et al. (2012). Models that demonstrate a negative correlation between population density and human fertility has been proposed (Lutz et al. 2007).

The concept of density is effective when dealing with an object that has a simple internal structure. By estimating the density of a region, we regard the region as having uniform density. However, it is necessary to be careful when dealing with complex social phenomena. Population density is normally obtained by dividing population by land area. However, the area inhabited by people is not the entire land area. We introduce definition for three types of land area to compare differences in population density.

We focus on population movement as it is easy to observe the effects on population density. Each person moves for various reasons, such as proceeding to the next stage of education, finding employment, or marriage. Regardless of reason, it is considered that the majority of people move from low population density places to high population density places. The result of these movements is the phenomenon of population agglomeration. Agglomeration effects have been modeled in several papers (Fujita and Ogawa 1982; Berliant and Konishi 2000; Ciccone 2002). This effect is related to various problems in modern society, such as population explosion in developing countries, aging in developed countries, and immigration issues. It is important to establish a method of analyzing agglomeration effects from empirical data.

In this study, we propose a method of analyzing agglomeration effects by observing relationship between population density and population increase or decrease. The data used for the analysis is census data and land use data for Japan. Population growth can be regarded as a function in terms of population density. We are then able to cleanly observe the phenomenon using integrated function because fluctuations in the original function are cancelled during integration.

When aggregating a population and an area, it is necessary to determine regional units for aggregation. Regional units can be determined using various methods of spatial division. The division method usually occurs at the administrative level, such as borders of prefectures. However, using administrative divisions is not optimal for reasons such as historical factors. For these reasons, various spatial division methods were proposed in Holmes and Lee (2009), Rozenfeld et al. (2011) and Fujimoto et al. (2015). We analyze agglomeration effects under both administrative spatial division and square block spatial division. The results obtained do not depend on the method of spatial division. A key point in obtaining the results is definition of land area for evaluating population density.

## 2 Details of the data

The Japanese Ministry of Land, Infrastructure, Transport and Tourism provides land use data on its website (http://nlftp.mlit.go.jp/ksj/). The land use data are also provided in a mesh data format. A mesh of about 100\(\times\)100 m\(^2\) provides the highest accuracy for land use. In this data, a land use code (see Table 1) is assigned to each cell.

We estimated three types of area by combining areas whose land use codes are checked in Table 1. Type I land is all land. This type is estimated by the land use codes 1, 2, 5, 6, 7, 9, A, B, E and G.

Type II land is inhabitable land. An inhabitable place is defined as any place that is fit for humans to live. Inhabitable land area can be estimated by subtracting such uninhabitable land as forests and lakes from the total land area. This type is estimated by land use codes 1, 2, 7, 9, A and G. Only about 33% of Japan’s land area is inhabitable because there is much mountainous area. This percentage is smaller than for European countries. For example, the percentages of inhabitable area in Germany, France, and the United Kingdom are 68, 71, and 88%, respectively (Fujimoto et al. 2015).

Land use code assignment

Code | Land use description | Type I | Type II | Type III |
---|---|---|---|---|

1 | Paddy fields | ✓ | ✓ | |

2 | Other agricultural land | ✓ | ✓ | |

5 | Forest | ✓ | ||

6 | Wasteland | ✓ | ||

7 | Land for building | ✓ | ✓ | ✓ |

9 | Trunk transportation land | ✓ | ✓ | ✓ |

A | Other land | ✓ | ✓ | ✓ |

B | Rivers and lakes | ✓ | ||

E | Beach | ✓ | ||

F | Body of seawater | |||

G | Golf course | ✓ | ✓ | |

0 | Outside of the analysis range |

## 3 Dividing the spatial region

To measure population and land area, it is necessary to determine how the measurement region is to be divided. There are several methods of dividing the land area of Japan into measurement regions. In this paper, we compare the results obtained using various spatial division methods.

The first spatial division uses the following eight regions based on traditional ways of combining several prefectures: Hokkaido, Tohoku, Kanto, Chubu, Kinki, Chugoku, Shikoku, and Kyushu. The average land area of these eight regions is about 47,246 km\(^2\).

The second spatial division uses the borders of the 47 prefectures. The average land area of the prefectures is about 8,041 km\(^2\).

The third spatial division uses the Shi-Ku-Cho-Son borders. Shi-Ku-Cho-Son are municipal units incorporated for local self-government. The closest English words to Shi, Ku, Cho and Son are city, ward, town and village, respectively. These borders have sometimes changed, such as in municipal mergers. We use the borders in 2010. The numbers of Shi, Ku, Cho and Son in 2010 were 767, 23, 757 and 184, respectively. The land area distribution of Shi-Ku-Cho-Son roughly obeys an exponential distribution. The average land area of Shi-Ku-Cho-Son is about 218 km\(^2\).

The fourth spatial division is into square blocks. In this case, we can control the scale of the spatial division by changing the size of the squares.

## 4 Correlation between population movement and population density

Let us introduce some variables for analyzing population movement.

*i*) indicates the region.

*i*) is given by

*i*) in 2005 is given by

*i*) in 2010 is given by

*i*) give totals for all of Japan. However, the reduction rate in this age group is about 2% (see Fig. 1). To reduce the influence of this reduction, we introduce normalized populations:

*i*) is given by

*i*) is given by

We perform the same analysis using square blocks. The gray dots in the bottom left figure in Fig. 2 indicate 15\(\times\)15 km\(^2\) square blocks used as the spatial division method; the crosses indicate 90\(\times\)90 km\(^2\) square blocks used as the spatial division method; the black circles indicate 200\(\times\)200 km\(^2\) square blocks used as the spatial division method. Note that the area of a 15\(\times\)15 km\(^2\) square is approximately the average area of a Shi-Ku-Cho-Son unit, the area of a 90\(\times\)90 km\(^2\) square is approximately the average area of a prefecture, and the area of a 200\(\times\)200 km\(^2\) square is approximately the average area of the traditional eight regions.

There are two important quantities that should be determined from the left figures in Fig. 2. The first is the whole slope, which represents the strength of the force moving 15–19 year-olds from low population density areas to high population density areas. The second quantity is the value of *x* where *y* is equal to zero (*x*-intercept). This value represents the boundary between population increase and population decrease.

*a*and

*b*with respect to

*y*-intercept and whole slope. We tried regression analysis by

*x*region. For this reason, regression analysis do not work well.

*x*-intercept not using regression analysis, we introduce the concept of integration. The relationship between a function

*f*(

*x*) acting at position

*x*and integrated function

*F*(

*x*) is

*x*as position and the vertical axis

*y*as a function, \(y=f(x)\). We implement the integration of Eq. (9) on the discrete scatter plot data as follows:

- 1.Sort the data in ascending order of
*x*:$$\begin{aligned} \cdots , (x^{(i)},y^{(i)}), (x^{(i+1)},y^{(i+1)}), \cdots \quad x^{(i)}<x^{(i+1)}. \end{aligned}$$(10) - 2.Evaluate the integrated function \(F(x^{(i)})\) using$$\begin{aligned} F(x^{(i)})=-\sum _{j=2}^i y^{(j)}(x^{(j-1)}-x^{(j)}). \end{aligned}$$(11)

The solid curve in the top right figure in Fig. 2 shows the integrated evaluated from the black circles in the top left figure. For easy comparison with results under other conditions, the maximum value of the integrated function is set to zero. The whole slope of the top left figure is expressed by the curvature of the curve. The position of *x* where \(y=0\) in the top left figure corresponds to the peak of the curve.

The solid gray curve in the top right figure in Fig. 2 shows the integrated function evaluation from the crosses in the top left figure. The dashed curve in the top right figure in Fig. 2 shows the integrated function evaluated from the gray dots in the top left figure. The dotted curve in the top right figure in Fig. 2 shows the integrated function evaluated using a 40\(\times\)40 km\(^2\) square block spatial division method for comparison.

The bottom right figure in Fig. 2 shows the same analysis as the top right figure for the square block spatial division method. The solid gray curve in the bottom right figure shows the integrated function evaluated from the gray dots in the bottom left figure; the dashed curve in the bottom right figure shows the integrated function evaluated from the crosses in the bottom left figure; the solid curve in the bottom right figure shows the integrated function evaluated from the black circles in the bottom left figure. Although the smoothness of the curve is lost, the results are almost same as for the administrative spatial division methods.

## 5 Comparison among the three types of definition of population density

The population density is obtained by dividing the population in a certain region by the land area of the region. It does not make sense to include uninhabitable land in the land area, where inhabitable places are defined as any place fit for humans to live. The total inhabitable area is calculated as the total type II area in Table 1.

Let us further reconsider population density. It appears that regions with land uses such as agriculture are unrelated to the movement of 15–19 years old. Type III land in Table 1 removes codes such as agriculture from type II land.

Figure 4 and Table 2 show the results using the same analysis as in the previous section, with the population density evaluated using type III land area rather than type I land area. Let us compare Figs. 4 and 3. The curvatures of the curves in Fig. 4 are sharper than those in Fig. 3. The differences between the peaks of the curves in Fig. 4 are smaller than the differences in Fig. 4. The results depend less on the spatial division method when using type III land area instead of type II land area. The population density at the peaks are \(10^{3.623}, 10^{3.726}\) and \(10^{3.782}\), when we use the spatial division 8-regions, prefectures and Shi-Ku-Cho-Son with respectively. The population density at the peaks are \(10^{3.682}, 10^{3.748}\) and \(10^{3.787}\), when we use the spatial division 200\(\times\)200, 90\(\times\)90 and 15\(\times\)15 \(\mathrm{km}^2\) squares with respectively. The population density at the peak is approximately \(10^{3.6}\)–\(10^{3.8}\sim\) 4,000–6,000\(/\mathrm{km}^2\).

*x*. It is considered the fluctuation closely around the critical point is important.

Type | Division | \(a\pm\) (error) | \(b\pm\) (error) | \(R^2\) | \(-a/b\) |
---|---|---|---|---|---|

I | 8 regions | −0.253 ± 0.092$$ | \(0.0889\pm 0.0037\) | 0.41 | 2.84 |

Prefectures | −0.345 ± 0.031 | \(0.116\pm 0.012\) | 0.67 | 2.97 | |

Shi-ku-sho-son | \(-0.3244\pm 0.083\) | \(0.0944\pm 0.0032\) | 0.32 | 3.43 | |

200\(\times\)200 | \(-0.250\pm 0.089\) | \(0.067\pm 0.042\) | 0.039 | 3.75 | |

90\(\times\)90 | \(-0.157\pm 0.041\) | \(0.021\pm 0.019\) | 0.0010 | 7.38 | |

15\(\times\)15 | \(-0.241\pm 0.010\) | \(0.0444\pm 0.0048\) | 0.039 | 5.42 | |

II | 8 regions | \(-0.32\pm 0.16\) | \(0.092\pm 0.052\) | 0.23 | 3.41 |

Prefectures | \(-0.540\pm 0.054\) | \(0.158\pm 0.017\) | 0.64 | 3.41 | |

Shi-ku-sho-son | \(-0.477\pm 0.015\) | \(0.1301\pm 0.0048\) | 0.28 | 3.67 | |

200\(\times\)200 | \(-0.28\pm 0.14\) | \(0.062\pm 0.050\) | 0.014 | 4.53 | |

90\(\times\)90 | \(-0.26\pm 0.11\) | \(0.059\pm 0.039\) | 0.011 | 4.42 | |

15\(\times\)15 | \(-0.283\pm 0.019\) | \(0.0480\pm 0.0070\) | 0.022 | 5.90 | |

III | 8 regions | \(-0.59\pm 0.33\) | \(0.157\pm 0.092\) | 0.21 | 3.77 |

Prefectures | \(-0.96\pm 0.12\) | \(0.256\pm 0.035\) | 0.53 | 3.76 | |

Shi-ku-sho-son | \(-0.710\pm 0.031\) | \(0.1866\pm 0.0088\) | 0.17 | 3.80 | |

200\(\times\)200 | \(-0.55\pm 0.32\) | \(0.129\pm 0.094\) | 0.022 | 4.24 | |

90\(\times\)90 | \(-0.44\pm 0.23\) | \(0.095\pm 0.068\) | 0.0097 | 4.61 | |

15\(\times\)15 | \(-0.230\pm 0.032\) | \(0.022\pm 0.010\) | 0.0019 | 10.2 |

Logarithmic population density of capital cities of prefectures

Prefecture | Capital | Density | Prefecture | Capital | Density |
---|---|---|---|---|---|

Hokkaido | Sapporo | 3.90 | Shiga | Otsu | 3.76 |

Aomori | Aomori | 3.51 | Kyoto | Kyoto | 4.14 |

Iwate | Morioka | 3.55 | Osaka | Osaka | 4.20 |

Miyagi | Sendai | 3.76 | Hyogo | Kobe | 3.94 |

Akita | Akita | 3.49 | Nara | Nara | 3.78 |

Yamagata | Yamagata | 3.55 | Wakayama | Wakayama | 3.74 |

Fukushima | Fukushima | 3.53 | Tottori | Tottori | 3.58 |

Ibaraki | Mito | 3.58 | Shimane | Matsue | 3.61 |

Tochigi | Utsunomiya | 3.56 | Okayama | Okayama | 3.72 |

Gumma | Maebashi | 3.60 | Hiroshima | Hiroshima | 4.03 |

Saitama | Saitama | 4.11 | Yamaguchi | Yamaguchi | 3.50 |

Chiba | Chiba | 3.72 | Tokushima | Tokushima | 3.78 |

Tokyo | Shinjuku | 4.27 | Kagawa | Takamatsu | 3.73 |

Kanagawa | Yokohama | 3.94 | Ehime | Matsuyama | 3.77 |

Niigata | Niigata | 3.78 | Kochi | Kochi | 3.78 |

Toyama | Toyama | 3.55 | Fukuoka | Fukuoka | 3.92 |

Ishikawa | Kanazawa | 3.71 | Saga | Saga | 3.61 |

Fukui | Fukui | 3.56 | Nagasaki | Nagasaki | 3.79 |

Yamanashi | Kofu | 3.69 | Kumamoto | Kumamoto | 3.76 |

Nagano | Nagano | 3.57 | Oita | Oita | 3.64 |

Gifu | Gifu | 3.73 | Miyazaki | Miyazaki | 3.66 |

Shizuoka | Shizuoka | 3.80 | Kagoshima | Kagoshima | 3.75 |

Aichi | Nagoya | 4.00 | Okinawa | Naha | 3.96 |

Mie | Tsu | 3.53 |

## 6 Analysis of various age classes

Clean peaks are not observed in the other classes. There is not much correlation between the logarithmic population density and the growth ratio of the population. For example, we show a scatter plot (see Fig. 7) of these two values for ages 20–24 in 2005.

## 7 Conclusion

We analyze the relationship between the population density and the growth ratio of the population. The relationship is clear for ages 10–14 in 2005 (ages 15–19 in 2010) and ages 15–19 in 2005 (ages 20–24 in 2010). This shows the agglomeration effect that people gather in more densely populated regions. The value of the population density that indicates a critical point in population growth is important. We evaluate this value using the concepts of integration. It is necessary to decide how to divide the land area when the population and the area are combined. We found that the results do not depend on how the area is divided when evaluating population density using the inhabitable area without agricultural land. It is concluded that around 4,000–6,000 /km\(^2\) population density is the critical point. Although the curve of integrated function is considered to be universal curve throughout Japan, the fluctuations from the curve depend on how to divide the land area.

We do not observe clear correlations between population density and population growth in the other age classes. Nevertheless, it is difficult to assume that people in these classes move randomly. It is possible that a stratified analysis using life events would be more useful than an analysis based on age group. Various life events, such as entering school, starting work, changing jobs, getting married, getting divorced, and recuperating can affect population movement. We would like to find quantities that relate to population movement independently of population density. For getting a job, we guess that per capita GDP density is more important than population density. For recuperation, we guess that the density of hospitals is more important than population density.

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